Quadratics

Standard Form, Vertex Form, Factored Form.

Quadratics

Standard Form, Vertex Form, Factored Form.

Analyzing quadratics

To determine whether a graph has a quadratic relation it must have an equal second difference. As you can see from the picture below by using the equation Y=x² the first difference is not the same, but when you calculate second difference they are all two. From this you can conclude that a quadratic relationship must have an equal second difference

Quadratic Terms

Example

From the graph below you can see the graph is pointing out the vertex, min/max value, the axis of symmetry, y-int and x-int.

Vertex - (1,9)

Min/max value- Y=9

Axis of symm - x=1

X-intercept - (-2,0) (4,0)

Y-intercept - (-8,0)

Vertex

Vertex Form

Y=A(X-H)²+K

In vertex form

The H in the equation is the axis of symmetry (x=h)

The K is the optimal value (y=k)

The A determines whether it will be stretched or compressed, and if its reflecting up or down

y=-1(x+3)²-2

As you can see The axis of symmetry is -3 because the h is +3. The Optimal value is -2.

And the parabola opens down cause of -1

Step pattern

The standard graph when graphing a parabola is Y=X² this is called the step pattern

Graphing Vertex Form

When using Vertex Form a simple way to graph Vertex form is to use the Y=X² graph and simply move the points. Y=2(x-2)²+2, from this equation you move your line right 2 because of the H and after that you move it 2 up because of your K. And for your x values from the y=x² would be multiplied by 2 to know how far the points are as seen in the graph. as you can see from the new graph represented by blue the line was shifted 2 to the right, 2 up and instead of the points being 1,4 they were 2,8

Completing the square (Changing from standard form to vertex form)

Y=2x²+12x-3

When solving this equation the first thing you want to do is remove a common factor

Y=2(x²+12x)-3

Find the constant that must be added and subtracted to create a perfect square(12/2)²

Y=2(x²+6x+9-9)-3

Group the 3 terms that form a perf square to do this multiple 2 by the -9 and move the bracket

Y=2(x²+6x+9)-18-3

Factor perfect square and collect like terms

Y=2(x+3)²-21

Graphing Vertex Form Part B

Y=-(X+5)²+1

With a problem like this first you want to find your y-intercept so you set your X to 0

Y=-(0+5)²+1

Then you just solve it like a normal equation

Y=-(5)²+1

Y=-25+1

Y=-24

Therefore your answer is (0,-24)

Graphing Vertex Form Part B

Y=-(X+5)²+1

Now that you have your y-intercept you must find your zeros, to do this set your y to zero

0=-(X+5)²+1

Firstly move k to the other side (1)

-1=-(x+5)²

Then you divide each side by your A (-1)

1=(X+5)²

After that you move your H (5)

-5+/-1=X

Lastly you solve for X

X=-5+1,X=-5-2

X=-4,X=-6

(-4,0)(-6,0)

Factored Form

Graphing Factored Form

Y=1(x+2)(x-4). When graphing Factored form you first must state your zeros. X+2=0, X=-2.

X-4=0, X=4. Then you must state the axis of symmetry by adding your x's and dividing by 2.

-2+4/2= 1.Once you have your axis of symmetry you sub it into your equation as you can see from the picture you have all the three points needed to complete your graph the vertex and the two x intercepts

Word problems

A mall has projected there revenue, represented by P, described by this equation P= -20(X-2)(X-16), when x represents the number of shoppers that arrive throughout the day.

A) How many shoppers does the mall need to have, so it will have a revenue of zero

When solving this question the first thing you want to do is refer to the graph. From the graph we can conclude that the mall needs to have 2 or 16 shoppers because thats where your x-intercepts are.

B) At how many shoppers does the mall make the most money?

Again when solving this question the first thing you want to do is look at the graph. You can clearly see that when there are 9 shoppers the mall makes the most money.

A shop is selling Pens for 20 each, 300 customers are willing to buy. For every $5 increase 30 fewer customers are willing to buy.

A) What is the maximum revenue the store can get?

First you want to put down a let statement

Let X represent the amount of $5 increase

Then you write out your equation which would be

R=(300-30X)(20+5X)

This is the equation because the first part is the number of customers which is 300 then 30 fewer which translates to 300-30

The second part is 20+5x because it represents price which is currently 30 and increases by 5

Now you substitute your y value and find X which is shown below.

Now that you have your x=ints you add them and divide by two

10-4/2= 3

Finally to find the max revenue your plug your 3 into the first equation

R=(300-30(3))(20+5(3))

R=(210)(35)

R=7450

Therefore the maximum revenue the store can make is 7450

How it relates to graphing parabolas

Expanding and factoring relate to graphing because with some equations you can factor or expand them to make graphing them easier and there is always a way to switch from vertex to factored to standard form

Quadratic Formula

To Find the X-intercept from standard form without factoring it you can use the quadratic formula shown before. X²+5X+6 When solving this you substitute in your A,B,C and solve as shown below. So the answer is (-6,0) (1,0)

The discriminant

The discriminant is found by a simple equation B²-4(a)(c). If the discriminant is in the negatives there are 0 solutions (x-intercepts) to the answer if its 0 then there is 1 solution (x-intercept) and if its positive then theres 2 solutions (x-intercepts)

Standard Form

Expanding

When expanding a problem you must multiply the number outside the bracket to the numbers in the brackets. As you can see from the picture when expanding you multiply the 5 to the 2x which gives you 10x, then you multiple it by the second term 5 times 1 and that equals 5. After that you put your terms together and the answer is 10x+5

Factoring

Common Factoring

To common factor all of your terms must be divisible by the same number. For example the picture to the right. From the picture you can see that first what you want to do is divide all your terms by 5 then put them in brackets and put a 5 in front of the brackets. Or you can put the 5 then put the brackets and add your terms. From the picture we can conclude that 15+10x+5 can be factored down to 5(3+2x+1)

Perfect Squares

On rare occasions you can factor a problem by squaring the first and last term. From the picture to the right we can see what factoring a perfect square looks like

Difference of squares

When factoring difference of squares you square the two terms again but this time instead of putting a square for the brackets you put them side by side to cancel out the middle term as seen in the exmaple

Simple trinomial factoring

My quiz and what i did wrong

Reflection

On this quiz i completely forgot about perfect squares and different squares and it showed. I lost a lot of marks on simple things like squaring the first and last term. I realized that i need to start learning which method of factoring i need to use to solve a specific problem. I hoped to better this understanding during the unit test but i forget that finding the x-intercepts you had to change standard form into factored form and lost a lot of marks. In the future to issue this problem i will study and review my notes more before hand.